where \(\theta _1(q,y)=I\sum _{r\in {Z}+\frac{1}{2}}(-1)^{r-\frac{1}{2}} q^{r^2/2} y^r\)  is the Jacobi theta series (the dots in the middle line indicate the obvious additional terms required for invariance under \(y\mapsto 1/y\) ).
Further examples of generating series of Gieseker indices of rank \(r>1\)  can be found in [1]}.
